Calculus early trans 8e james stewart from www mathschoolinternational com

Edward Dobson, Mississippi State University Isaac Goldbring, University of Illinois at Chicago Lea Jenkins, Clemson UniversityRebecca Wahl, Butler University Maria Andersen, Muskegon Com

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PrEfacE xi

To ThE sTudEnT xxiii calculaTors, comPuTErs, and oThEr graPhing dEvicEs xxiv diagnosTic TEsTs xxvi

1

1.1 Four Ways to Represent a Function 10

1.2 Mathematical Models: A Catalog of Essential Functions 23

1.3 New Functions from Old Functions 36

2.1 The Tangent and Velocity Problems 78

2.2 The Limit of a Function 83

2.3 Calculating Limits Using the Limit Laws 95

2.4 The Precise Definition of a Limit 104

2.5 Continuity 114

2.6 Limits at Infinity; Horizontal Asymptotes 126

2.7 Derivatives and Rates of Change 140

2.8 The Derivative as a Function 152Review 165

Problems Plus 169

Contents

3.1 Derivatives of Polynomials and Exponential Functions 172

3.2 The Product and Quotient Rules 183

3.3 Derivatives of Trigonometric Functions 190

3.4 The Chain Rule 197

3.5 Implicit Differentiation 208

3.6 Derivatives of Logarithmic Functions 218

3.7 Rates of Change in the Natural and Social Sciences 224

3.8 Exponential Growth and Decay 237

3.9 Related Rates 245

3.10 Linear Approximations and Differentials 251

3.11 Hyperbolic Functions 259 Review 266

Problems Plus 270

4

4.2 The Mean Value Theorem 287

4.3 How Derivatives Affect the Shape of a Graph 293

4.4 Indeterminate Forms and l’Hospital’s Rule 304

4.5 Summary of Curve Sketching 315

4.6 Graphing with Calculus and Calculators 323

4.7 Optimization Problems 330

4.9 Antiderivatives 350 Review 358

Problems Plus 363

5.1 Areas and Distances 366

5.2 The Definite Integral 378

5.3 The Fundamental Theorem of Calculus 392

5.4 Indefinite Integrals and the Net Change Theorem 402

5.5 The Substitution Rule 412 Review 421

Problems Plus 425

6

6.1 Areas Between Curves 428

6.3 Volumes by Cylindrical Shells 449

6.5 Average Value of a Function 461

7.4 Integration of Rational Functions by Partial Fractions 493

7.5 Strategy for Integration 503

7.6 Integration Using Tables and Computer Algebra Systems 508

7.7 Approximate Integration 514

7.8 Improper Integrals 527 Review 537

Problems Plus 540

8.2 Area of a Surface of Revolution 551

8.3 Applications to Physics and Engineering 558

8.4 Applications to Economics and Biology 569

8.5 Probability 573 Review 581

Problems Plus 583

9

9.1 Modeling with Differential Equations 586

9.2 Direction Fields and Euler’s Method 591

9.3 Separable Equations 599

9.4 Models for Population Growth 610

9.5 Linear Equations 620

9.6 Predator-Prey Systems 627 Review 634

Problems Plus 637

10

10.1 Curves Defined by Parametric Equations 640

10.2 Calculus with Parametric Curves 649

10.3 Polar Coordinates 658

10.4 Areas and Lengths in Polar Coordinates 669

11.3 The Integral Test and Estimates of Sums 719

11.4 The Comparison Tests 727

11.5 Alternating Series 732

11.6 Absolute Convergence and the Ratio and Root Tests 737

11.7 Strategy for Testing Series 744

11.8 Power Series 746

11.9 Representations of Functions as Power Series 752

11.10 Taylor and Maclaurin Series 759

11.11 Applications of Taylor Polynomials 774

12.3 The Dot Product 807

12.4 The Cross Product 814

12.5 Equations of Lines and Planes 823

12.6 Cylinders and Quadric Surfaces 834

Review 841

Problems Plus 844

13.1 Vector Functions and Space Curves 848

13.2 Derivatives and Integrals of Vector Functions 855

13.3 Arc Length and Curvature 861

13.4 Motion in Space: Velocity and Acceleration 870

Review 881

Problems Plus 884

14

14.1 Functions of Several Variables 888

14.2 Limits and Continuity 903

14.3 Partial Derivatives 911

14.4 Tangent Planes and Linear Approximations 927

14.5 The Chain Rule 937

14.6 Directional Derivatives and the Gradient Vector 946

14.8 Lagrange Multipliers 971

Review 981

Problems Plus 985

15

15.1 Double Integrals over Rectangles 988

15.2 Double Integrals over General Regions 1001

15.3 Double Integrals in Polar Coordinates 1010

15.4 Applications of Double Integrals 1016

15.5 Surface Area 1026

15.6 Triple Integrals 1029

15.7 Triple Integrals in Cylindrical Coordinates 1040

15.8 Triple Integrals in Spherical Coordinates 1045

15.9 Change of Variables in Multiple Integrals 1052Review 1061

16.5 Curl and Divergence 1103

16.6 Parametric Surfaces and Their Areas 1111

16.7 Surface Integrals 1122

16.8 Stokes’ Theorem 1134

16.9 The Divergence Theorem 1141

Review 1148

Problems Plus 1151

17

17.1 Second-Order Linear Equations 1154

17.2 Nonhomogeneous Linear Equations 1160

17.3 Applications of Second-Order Differential Equations 1168

17.4 Series Solutions 1176

Review 1181

A Numbers, Inequalities, and Absolute Values A2

B Coordinate Geometry and Lines A10

C Graphs of Second-Degree Equations A16

D Trigonometry A24

E Sigma Notation A34

F Proofs of Theorems A39

G The Logarithm Defined as an Integral A50

I Answers to Odd-Numbered Exercises A65

The art of teaching, Mark Van Doren said, is the art of assisting discovery I have tried

to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty In this edition, as in the first seven editions, I aim to convey

to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject Newton undoubtedly experienced a sense of triumph when he made his great discoveries I want students to share some of that excitement

The emphasis is on understanding concepts I think that nearly everybody agrees that this should be the primary goal of calculus instruction In fact, the impetus for the cur-rent calculus reform movement came from the Tulane Conference in 1986, which for-mulated as their first recommendation:

Focus on conceptual understanding.

I have tried to implement this goal through the Rule of Three: “Topics should be

pre-sented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach concep-tual reasoning in fundamental ways More recently, the Rule of Three has been expanded

to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as

well

In writing the eighth edition my premise has been that it is possible to achieve ceptual understanding and still retain the best traditions of traditional calculus The book contains elements of reform, but within the context of a traditional curriculum

con-I have written several other calculus textbooks that might be preferable for some tors Most of them also come in single variable and multivariable versions

instruc-● Calculus, Eighth Edition, is similar to the present textbook except that the

exponen-tial, logarithmic, and inverse trigonometric functions are covered in the second semester

● Essential Calculus, Second Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Eighth Edition The relative brevity is

achieved through briefer exposition of some topics and putting some features on the website

● Essential Calculus: Early Transcendentals, Second Edition, resembles Essential

Calculus, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

G E O R G E P O LYA

Preface

● Calculus: Concepts and Contexts, Fourth Edition, emphasizes conceptual

under-standing even more strongly than this book The coverage of topics is not pedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters

encyclo-● Calculus: Early Vectors introduces vectors and vector functions in the first semester

and integrates them throughout the book It is suitable for students taking ing and physics courses concurrently with calculus

engineer-● Brief Applied Calculus is intended for students in business, the social sciences, and

the life sciences

● Biocalculus: Calculus for the Life Sciences is intended to show students in the life

sciences how calculus relates to biology

● Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three addi-

tional chapters covering probability and statistics

The changes have resulted from talking with my colleagues and students at the sity of Toronto and from reading journals, as well as suggestions from users and review-ers Here are some of the many improvements that I’ve incorporated into this edition:

Univer-● The data in examples and exercises have been updated to be more timely

● New examples have been added (see Examples 6.1.5, 11.2.5, and 14.3.3, for instance) And the solutions to some of the existing examples have been amplified

● Three new projects have been added: The project Controlling Red Blood Cell Loss

During Surgery (page 244) describes the ANH procedure, in which blood is extracted from the patient before an operation and is replaced by saline solution This dilutes the patient’s blood so that fewer red blood cells are lost during bleed-ing and the extracted blood is returned to the patient after surgery The project

Planes and Birds: Minimizing Energy (page 344) asks how birds can minimize

power and energy by flapping their wings versus gliding In the project The Speedo

LZR Racer (page 936) it is explained that this suit reduces drag in the water and, as

a result, many swimming records were broken Students are asked why a small decrease in drag can have a big effect on performance

● I have streamlined Chapter 15 (Multiple Integrals) by combining the first two tions so that iterated integrals are treated earlier

sec-● More than 20% of the exercises in each chapter are new Here are some of my favorites: 2.7.61, 2.8.36–38, 3.1.79–80, 3.11.54, 4.1.69, 4.3.34, 4.3.66, 4.4.80, 4.7.39, 4.7.67, 5.1.19–20, 5.2.67–68, 5.4.70, 6.1.51, 8.1.39, 12.5.81, 12.6.29–30, 14.6.65–66 In addition, there are some good new Problems Plus (See Problems 12–14 on page 272, Problem 13 on page 363, Problems 16–17 on page 426, and Problem 8 on page 986.)

conceptual Exercises

The most important way to foster conceptual understanding is through the problems that we assign To that end I have devised various types of problems Some exercise sets begin with requests to explain the meanings of the basic concepts of the section (See, for instance, the irst few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35–38, 2.8.47–52, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38, 14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and 16.3.1–2)

Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.10, 2.8.66, 4.3.69–70, and 7.8.67) I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.45–46, 3.7.27, and 9.4.4)

graded Exercise sets

Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs

real-world data

My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data to introduce, motivate, and illustrate the concepts of calculus As a result, many of the examples and exercises deal with functions deined by such numerical data or graphs

See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.35 (unemployment rates), Exercise 5.1.16 (velocity of the space shuttle

Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption) Functions

of two variables are illustrated by a table of values of the wind-chill index as a function

of air temperature and wind speed (Example 14.1.2) Partial derivatives are introduced

in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity This example is pursued further in connection with linear approximations (Example 14.4.3)

Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas

Double integrals are used to estimate the average snowfall in Colorado on December 20–21, 2006 (Example 15.1.9) Vector ields are introduced in Section 16.1 by depictions

of actual velocity vector ields showing San Francisco Bay wind patterns

Projects

One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplish-

ment when completed I have included four kinds of projects: Applied Projects involve

applications that are designed to appeal to the imagination of students The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height

or to fall back to its original height (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of

a rocket so as to minimize the total mass while enabling the rocket to reach a desired

velocity Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer Writ-

ing Projects ask students to compare present-day methods with those of the founders of

calculus—Fermat’s method for inding tangents, for instance Suggested references are

supplied Discovery Projects anticipate results to be discussed later or encourage

dis-covery through pattern recognition (see the one following Section 7.6) Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6), and intersections of three cylinders (after Section 15.7) Additional projects can be found

in the Instructor’s Guide (see, for instance, Group Exercise 5.1: Position from Samples).

Problem solving

Students usually have dificulties with problems for which there is no single well-deined procedure for obtaining the answer I think nobody has improved very much on George Polya’s four-stage problem-solving strategy and, accordingly, I have included a version

of his problem-solving principles following Chapter 1 They are applied, both explicitly and implicitly, throughout the book After the other chapters I have placed sections called

Problems Plus, which feature examples of how to tackle challenging calculus problems

In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: “A mathematical problem should be dificult in order to entice us, yet not inaccessible lest it mock our efforts.” When I put these challenging problems on assignments and tests I grade them in a different way Here I reward a student signii-cantly for ideas toward a solution and for recognizing which problem-solving principles are relevant

Technology

The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen But, when properly used, graphing calculators and computers are powerful tools for discovering and understand-ing those concepts This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required The icon ; indicates an exercise that deinitely requires the use of such technology, but that is not to say that it can’t be used on the other exercises as well The symbol CAS

is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89) are required But technology doesn’t make pencil and paper obsolete Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate

Tools for Enriching calculus

TEC is a companion to the text and is intended to enrich and complement its contents (It is now accessible in the eBook via CourseMate and Enhanced WebAssign Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC Modules that provide a laboratory environment in which

they can explore the topic in different ways and at different levels Visuals are tions of igures in text; Modules are more elaborate activities and include exercises

anima-Instructors can choose to become involved at several different levels, ranging from ply encouraging students to use the Visuals and Modules for independent exploration,

sim-to assigning speciic exercises from those included with each Module, or sim-to creating additional exercises, labs, and projects that make use of the Visuals and Modules

TEC also includes Homework Hints for representative exercises (usually bered) in every section of the text, indicated by printing the exercise number in red

odd-num-These hints are usually presented in the form of questions and try to imitate an effective teaching assistant by functioning as a silent tutor They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress

The system also includes Active Examples, in which students are guided in step tutorials through text examples, with links to the textbook and to video solutions

step-by-website

Visit CengageBrain.com or stewartcalculus.com for these additional materials:

● Homework Hints

● Algebra Review

● Lies My Calculator and Computer Told Me

● History of Mathematics, with links to the better historical websites

● Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes

● Archived Problems (Drill exercises that appeared in previous editions, together with their solutions)

● Challenge Problems (some from the Problems Plus sections from prior editions)

● Links, for particular topics, to outside Web resources

● Selected Visuals and Modules from Tools for Enriching Calculus (TEC)

The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, tions, and Trigonometry

Func-This is an overview of the subject and includes a list of questions to motivate the study

of calculus

From the beginning, multiple representations of functions are stressed: verbal, cal, visual, and algebraic A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view

numeri-The material on limits is motivated by a prior discussion of the tangent and velocity problems Limits are treated from descriptive, graphical, numerical, and algebraic points

of view Section 2.4, on the precise definition of a limit, is an optional section Sections

diagnostic Tests

a Preview of calculus

1 functions and models

2 limits and derivatives

2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3 Here the exam-ples and exercises explore the meanings of derivatives in various contexts Higher deriva-tives are introduced in Section 2.8.

All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here When derivatives are computed in applied situations, students are asked to explain their meanings Exponential growth and decay are now covered in this chapter

The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves Some substantial optimi-zation problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow

The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables

Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration General methods are emphasized The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral

All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation Accordingly, in Section 7.5,

I present a strategy for integration The use of computer algebra systems is discussed in Section 7.6

Here are the applications of integration—arc length and surface area—for which it is useful to have available all the techniques of integration, as well as applications to biol-ogy, economics, and physics (hydrostatic force and centers of mass) I have also included a section on probability There are more applications here than can realistically

be covered in a given course Instructors should select applications suitable for their students and for which they themselves have enthusiasm

Modeling is the theme that unifies this introductory treatment of differential equations Direction fields and Euler’s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal consideration These methods are applied to the exponential, logistic, and other models for population growth The first four or five sections of this chapter serve as a good introduction to first-order differential equations An optional final section uses predator-prey models to illustrate systems of differential equations

This chapter introduces parametric and polar curves and applies the methods of calculus

to them Parametric curves are well suited to laboratory projects; the two presented here involve families of curves and Bézier curves A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13

The convergence tests have intuitive justifications (see page 719) as well as formal proofs Numerical estimates of sums of series are based on which test was used to prove convergence The emphasis is on Taylor series and polynomials and their applications

to physics Error estimates include those from graphing devices

The material on three-dimensional analytic geometry and vectors is divided into two chapters Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces

This chapter covers vector-valued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culmi-nating in Kepler’s laws

Functions of two or more variables are studied from verbal, numerical, visual, and braic points of view In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function

alge-of the actual temperature and the relative humidity

Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals

Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized

Since first-order differential equations are covered in Chapter 9, this final chapter deals with second-order linear differential equations, their application to vibrating springs and electric circuits, and series solutions

Calculus, Early Transcendentals, Eighth Edition, is supported by a complete set of ancillaries developed under my direction Each piece has been designed to enhance student understanding and to facilitate creative instruction The tables on pages xxi–xxii describe each of these ancillaries

The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers

I greatly appreciate the time they spent to understand my motivation for the approach taken I have learned something from each of them

Eighth Edition reviewers

Jay Abramson, Arizona State University Adam Bowers, University of California San Diego Neena Chopra, The Pennsylvania State University

11 ininite sequences and series

12 vectors and the

Edward Dobson, Mississippi State University Isaac Goldbring, University of Illinois at Chicago Lea Jenkins, Clemson University

Rebecca Wahl, Butler University

Maria Andersen, Muskegon Community College

Eric Aurand, Eastield College

Joy Becker, University of Wisconsin–Stout

Przemyslaw Bogacki, Old Dominion University

Amy Elizabeth Bowman, University of Alabama

in Huntsville

Monica Brown, University of Missouri–St Louis

Roxanne Byrne, University of Colorado at Denver and

Health Sciences Center

Teri Christiansen, University of Missouri–Columbia

Bobby Dale Daniel, Lamar University

Jennifer Daniel, Lamar University

Andras Domokos, California State University, Sacramento

Timothy Flaherty, Carnegie Mellon University

Lee Gibson, University of Louisville

Jane Golden, Hillsborough Community College

Semion Gutman, University of Oklahoma

Diane Hoffoss, University of San Diego

Lorraine Hughes, Mississippi State University

Jay Jahangiri, Kent State University

John Jernigan, Community College of Philadelphia

Brian karasek, South Mountain Community College Jason kozinski, University of Florida

Carole krueger, The University of Texas at Arlington ken kubota, University of Kentucky

John Mitchell, Clark College Donald Paul, Tulsa Community College Chad Pierson, University of Minnesota, Duluth Lanita Presson, University of Alabama in Huntsville karin Reinhold, State University of New York at Albany Thomas Riedel, University of Louisville

Christopher Schroeder, Morehead State University Angela Sharp, University of Minnesota, Duluth Patricia Shaw, Mississippi State University Carl Spitznagel, John Carroll University Mohammad Tabanjeh, Virginia State University Capt koichi Takagi, United States Naval Academy Lorna TenEyck, Chemeketa Community College Roger Werbylo, Pima Community College David Williams, Clayton State University Zhuan Ye, Northern Illinois University

Technology reviewers

B D Aggarwala, University of Calgary

John Alberghini, Manchester Community College

Michael Albert, Carnegie-Mellon University

Daniel Anderson, University of Iowa

Amy Austin, Texas A&M University

Donna J Bailey, Northeast Missouri State University

Wayne Barber, Chemeketa Community College

Marilyn Belkin, Villanova University

Neil Berger, University of Illinois, Chicago

David Berman, University of New Orleans

Anthony J Bevelacqua, University of North Dakota

Richard Biggs, University of Western Ontario

Robert Blumenthal, Oglethorpe University

Martina Bode, Northwestern University

Barbara Bohannon, Hofstra University

Jay Bourland, Colorado State University

Philip L Bowers, Florida State University

Amy Elizabeth Bowman, University of Alabama in Huntsville

Stephen W Brady, Wichita State University

Michael Breen, Tennessee Technological University

Robert N Bryan, University of Western Ontario

David Buchthal, University of Akron Jenna Carpenter, Louisiana Tech University Jorge Cassio, Miami-Dade Community College Jack Ceder, University of California, Santa Barbara Scott Chapman, Trinity University

Zhen-Qing Chen, University of Washington—Seattle James Choike, Oklahoma State University

Barbara Cortzen, DePaul University Carl Cowen, Purdue University Philip S Crooke, Vanderbilt University Charles N Curtis, Missouri Southern State College Daniel Cyphert, Armstrong State College

Robert Dahlin

M Hilary Davies, University of Alaska Anchorage Gregory J Davis, University of Wisconsin–Green Bay Elias Deeba, University of Houston–Downtown Daniel DiMaria, Suffolk Community College Seymour Ditor, University of Western Ontario Greg Dresden, Washington and Lee University Daniel Drucker, Wayne State University kenn Dunn, Dalhousie University

Previous Edition reviewers

Dennis Dunninger, Michigan State University

Bruce Edwards, University of Florida

David Ellis, San Francisco State University

John Ellison, Grove City College

Martin Erickson, Truman State University

Garret Etgen, University of Houston

Theodore G Faticoni, Fordham University

Laurene V Fausett, Georgia Southern University

Norman Feldman, Sonoma State University

Le Baron O Ferguson, University of California—Riverside

Newman Fisher, San Francisco State University

José D Flores, The University of South Dakota

William Francis, Michigan Technological University

James T Franklin, Valencia Community College, East

Stanley Friedlander, Bronx Community College

Patrick Gallagher, Columbia University–New York

Paul Garrett, University of Minnesota–Minneapolis

Frederick Gass, Miami University of Ohio

Bruce Gilligan, University of Regina

Matthias k Gobbert, University of Maryland, Baltimore County

Gerald Goff, Oklahoma State University

Stuart Goldenberg, California Polytechnic State University

John A Graham, Buckingham Browne & Nichols School

Richard Grassl, University of New Mexico

Michael Gregory, University of North Dakota

Charles Groetsch, University of Cincinnati

Paul Triantailos Hadavas, Armstrong Atlantic State University

Salim M Hạdar, Grand Valley State University

D W Hall, Michigan State University

Robert L Hall, University of Wisconsin–Milwaukee

Howard B Hamilton, California State University, Sacramento

Darel Hardy, Colorado State University

Shari Harris, John Wood Community College

Gary W Harrison, College of Charleston

Melvin Hausner, New York University/Courant Institute

Curtis Herink, Mercer University

Russell Herman, University of North Carolina at Wilmington

Allen Hesse, Rochester Community College

Randall R Holmes, Auburn University

James F Hurley, University of Connecticut

Amer Iqbal, University of Washington—Seattle

Matthew A Isom, Arizona State University

Gerald Janusz, University of Illinois at Urbana-Champaign

John H Jenkins, Embry-Riddle Aeronautical University,

Prescott Campus

Clement Jeske, University of Wisconsin, Platteville

Carl Jockusch, University of Illinois at Urbana-Champaign

Jan E H Johansson, University of Vermont

Jerry Johnson, Oklahoma State University

Zsuzsanna M kadas, St Michael’s College

Nets katz, Indiana University Bloomington

Matt kaufman

Matthias kawski, Arizona State University

Frederick W keene, Pasadena City College

Robert L kelley, University of Miami

Akhtar khan, Rochester Institute of Technology

Marianne korten, Kansas State University Virgil kowalik, Texas A&I University kevin kreider, University of Akron Leonard krop, DePaul University Mark krusemeyer, Carleton College John C Lawlor, University of Vermont Christopher C Leary, State University of New York at Geneseo David Leeming, University of Victoria

Sam Lesseig, Northeast Missouri State University Phil Locke, University of Maine

Joyce Longman, Villanova University Joan McCarter, Arizona State University Phil McCartney, Northern Kentucky University Igor Malyshev, San Jose State University Larry Mansield, Queens College Mary Martin, Colgate University Nathaniel F G Martin, University of Virginia Gerald Y Matsumoto, American River College James Mckinney, California State Polytechnic University, Pomona Tom Metzger, University of Pittsburgh

Richard Millspaugh, University of North Dakota Lon H Mitchell, Virginia Commonwealth University Michael Montađo, Riverside Community College Teri Jo Murphy, University of Oklahoma Martin Nakashima, California State Polytechnic University,

Pomona

Ho kuen Ng, San Jose State University Richard Nowakowski, Dalhousie University Hussain S Nur, California State University, Fresno Norma Ortiz-Robinson, Virginia Commonwealth University Wayne N Palmer, Utica College

Vincent Panico, University of the Paciic

F J Papp, University of Michigan–Dearborn Mike Penna, Indiana University–Purdue University Indianapolis Mark Pinsky, Northwestern University

Lothar Redlin, The Pennsylvania State University Joel W Robbin, University of Wisconsin–Madison Lila Roberts, Georgia College and State University

E Arthur Robinson, Jr., The George Washington University Richard Rockwell, Paciic Union College

Rob Root, Lafayette College Richard Ruedemann, Arizona State University David Ryeburn, Simon Fraser University Richard St Andre, Central Michigan University Ricardo Salinas, San Antonio College

Robert Schmidt, South Dakota State University Eric Schreiner, Western Michigan University Mihr J Shah, Kent State University–Trumbull Qin Sheng, Baylor University

Theodore Shifrin, University of Georgia Wayne Skrapek, University of Saskatchewan Larry Small, Los Angeles Pierce College Teresa Morgan Smith, Blinn College William Smith, University of North Carolina Donald W Solomon, University of Wisconsin–Milwaukee Edward Spitznagel, Washington University

Joseph Stampli, Indiana University

kristin Stoley, Blinn College

M B Tavakoli, Chaffey College

Magdalena Toda, Texas Tech University

Ruth Trygstad, Salt Lake Community College

Paul Xavier Uhlig, St Mary’s University, San Antonio

Stan Ver Nooy, University of Oregon

Andrei Verona, California State University–Los Angeles

klaus Volpert, Villanova University

Russell C Walker, Carnegie Mellon University

William L Walton, McCallie School

Peiyong Wang, Wayne State University Jack Weiner, University of Guelph Alan Weinstein, University of California, Berkeley Theodore W Wilcox, Rochester Institute of Technology Steven Willard, University of Alberta

Robert Wilson, University of Wisconsin–Madison Jerome Wolbert, University of Michigan–Ann Arbor Dennis H Wortman, University of Massachusetts, Boston Mary Wright, Southern Illinois University–Carbondale Paul M Wright, Austin Community College

Xian Wu, University of South Carolina

In addition, I would like to thank R B Burckel, Bruce Colletti, David Behrman, John Dersch, Gove Efinger, Bill Emerson, Dan kalman, Quyan khan, Alfonso Gracia-Saz, Allan MacIsaac, Tami Martin, Monica Nitsche, Lamia Raffo, Norton Starr, and Jim Trefz- ger for their suggestions; Al Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; George Bergman, David Bleecker, Dan Clegg, Victor kaftal, Anthony Lam, Jamie Lawson, Ira Rosen-holtz, Paul Sally, Lowell Smylie, and Larry Wallen for ideas for exercises; Dan Drucker for the roller derby project; Thomas Banchoff, Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, Philip Strafin, and klaus Volpert for ideas for projects; Dan Anderson, Dan Clegg, Jeff Cole, Dan Drucker, and Barbara Frank for solving the new exercises and suggesting ways to improve them; Marv Riedesel and Mary Johnson for accuracy in proofreading; Andy Bulman-Fleming, Lothar Redlin, Gina Sanders, and Saleem Watson for additional proofreading; and Jeff Cole and Dan Clegg for their careful preparation and proofreading of the answer manuscript

In addition, I thank those who have contributed to past editions: Ed Barbeau, George Bergman, Fred Brauer, Andy Bulman-Fleming, Bob Burton, David Cusick, Tom DiCic-cio, Garret Etgen, Chris Fisher, Leon Gerber, Stuart Goldenberg, Arnold Good, Gene Hecht, Harvey keynes, E L koh, Zdislav kovarik, kevin kreider, Emile LeBlanc, David Leep, Gerald Leibowitz, Larry Peterson, Mary Pugh, Lothar Redlin, Carl Riehm, John Ringland, Peter Rosenthal, Dusty Sabo, Doug Shaw, Dan Silver, Simon Smith, Saleem Watson, Alan Weinstein, and Gail Wolkowicz

I also thank kathi Townes, Stephanie kuhns, kristina Elliott, and kira Abdallah of TECHarts for their production services and the following Cengage Learning staff: Cheryll Linthicum, content project manager; Stacy Green, senior content developer; Samantha Lugtu, associate content developer; Stephanie kreuz, product assistant; Lynh Pham, media developer; Ryan Ahern, marketing manager; and Vernon Boes, art director They have all done an outstanding job

I have been very fortunate to have worked with some of the best mathematics editors

in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hayhurst, Gary Ostedt, Bob Pirtle, Richard Stratton, Liz Covello, and now Neha Taleja All of them have contributed greatly to the success of this book

james stewart

Each section of the text is discussed from several viewpoints

The Instructor’s Guide contains suggested time to allot, points

to stress, text discussion topics, core materials for lecture,

workshop/discussion suggestions, group work exercises in

a form suitable for handout, and suggested homework

assignments.

Complete Solutions Manual

Single Variable Early Transcendentals

By Daniel Anderson, Jeffery A Cole, and Daniel Drucker

ISBN 978-1-305-27239-2

Multivariable

By Dan Clegg and Barbara Frank

ISBN 978-1-305-27611-6

Includes worked-out solutions to all exercises in the text.

Printed Test Bank

By William Steven Harmon

This lexible online system allows you to author, edit, and

manage test bank content from multiple Cengage Learning

solutions; create multiple test versions in an instant; and

deliver tests from your LMS, your classroom, or wherever you

want.

Stewart Website

www.stewartcalculus.com

Contents: Homework Hints n Algebra Review n Additional

is accessible in the eBook via CourseMate and Enhanced WebAssign Selected Visuals and Modules are available at

www.stewartcalculus.com.

Enhanced WebAssign®

www.webassign.net Printed Access Code: ISBN 978-1-285-85826-5 Instant Access Code ISBN: 978-1-285-85825-8

Exclusively from Cengage Learning, Enhanced WebAssign offers an extensive online program for Stewart’s Calculus

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content both at the start of the course and at the beginning

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n Read It eBook pages, Watch It videos, Master It tutorials,

and Chat About It links

note-taking, and search features, as well as links to multimedia resources

n Personal Study Plans (based on diagnostic quizzing) that

identify chapter topics that students will need to master

equivalent mathematical responses in the same way an instructor grades

of seeing students’ detailed solutions

n Visualizing Calculus Animations, Lecture Videos, and more

YouBook is an eBook that is both interactive and

customiz-able Containing all the content from Stewart’s Calculus,

YouBook features a text edit tool that allows instructors to

modify the textbook narrative as needed With YouBook,

instructors can quickly reorder entire sections and chapters

or hide any content they don’t teach to create an eBook that

perfectly matches their syllabus Instructors can further

customize the text by adding instructor-created or YouTube

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YouBook is available within Enhanced WebAssign.

CourseMate

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Course-Mate for Stewart’s Calculus includes an interactive eBook,

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Student Solutions Manual

Single Variable Early Transcendentals

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ISBN 978-1-305-27242-2

Multivariable

By Dan Clegg and Barbara Frank

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Provides completely worked-out solutions to all odd-

numbered exercises in the text, giving students a chance to

to arrive at the answer The Student Solutions Manual can be ordered or accessed online as an eBook at

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Study Guide Single Variable Early Transcendentals

By Richard St Andre ISBN 978-1-305-27914-8

Multivariable

By Richard St Andre ISBN 978-1-305-27184-5

For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, and summary and focus questions with explained answers The Study Guide also contains “Technology Plus” questions and multiple-choice “On Your Own” exam-style questions The Study Guide can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN.

It is designed for calculus courses that integrate the review of precalculus concepts or for individual use Order a copy of the text or access the eBook online at www.cengagebrain.com

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Linear Algebra for Calculus

by konrad J Heuvers, William P Francis, John H kuisti, Deborah F Lockhart, Daniel S Moak, and Gene M Ortner ISBN 978-0-534-25248-9

This comprehensive book, designed to supplement the lus course, provides an introduction to and review of the basic ideas of linear algebra Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN.

calcu-■ Electronic items ■ Printed items

xxii

To the Student

Reading a calculus textbook is different from reading a

newspaper or a novel, or even a physics book Don’t be

dis-couraged if you have to read a passage more than once

in order to understand it You should have pencil and paper

and calculator at hand to sketch a diagram or make a

calculation

Some students start by trying their homework problems

and read the text only if they get stuck on an exercise I

sug-gest that a far better plan is to read and understand a section

of the text before attempting the exercises In particular, you

should look at the definitions to see the exact meanings of

the terms And before you read each example, I suggest that

you cover up the solution and try solving the problem

your-self You’ll get a lot more from looking at the solution if

you do so

Part of the aim of this course is to train you to think

logi-cally Learn to write the solutions of the exercises in a

con-nected, step-by-step fashion with explanatory sentences—

not just a string of disconnected equations or formulas

The answers to the odd-numbered exercises appear at the

back of the book, in Appendix I Some exercises ask for a

verbal explanation or interpretation or description In such

cases there is no single correct way of expressing the

answer, so don’t worry that you haven’t found the definitive

answer In addition, there are often several different forms

in which to express a numerical or algebraic answer, so if

your answer differs from mine, don’t immediately assume

you’re wrong For example, if the answer given in the back

of the book is s2 2 1 and you obtain 1y(1 1s2), then

you’re right and rationalizing the denominator will show

that the answers are equivalent

The icon ; indicates an exercise that definitely requires

the use of either a graphing calculator or a computer with

graphing software But that doesn’t mean that graphing

devices can’t be used to check your work on the other

exer-cises as well The symbol CAS is reserved for problems in

which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89) are required

You will also encounter the symbol |, which warns you against committing an error I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake

Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can

be accessed in the eBook via Enhanced WebAssign and CourseMate (selected Visuals and Modules are available at www.stewartcalculus.com) It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful

You will notice that some exercise numbers are printed

in red: 5. This indicates that Homework Hints are available

for the exercise These hints can be found on lus.com as well as Enhanced WebAssign and CourseMate The homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer You need to pursue each hint in an active manner with pencil and paper to work out the details If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint

stewartcalcu-I recommend that you keep this book for reference poses after you finish the course Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer

pur-Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect I hope you will discover that it is not only useful but also intrinsically beautiful

J A M E S S T E W A R T

Advances in technology continue to bring a wider variety of tools for doing mathematics Handheld calculators are becoming more pow-erful, as are software programs and Internet resources In addition, many mathematical applications have been released for smartphones and tablets such as the iPad

Some exercises in this text are marked with a graphing icon ;, which indicates that the use of some technology is required Often this means that we intend for a graphing device to be used in drawing the graph of a function or equation You might also need technology to ind the zeros of a graph or the points of intersection of two graphs

In some cases we will use a calculating device to solve an equation or evaluate a deinite integral numerically Many scientiic and graphing calculators have these features built in, such as the Texas Instruments TI-84 or TI-Nspire CX Similar calculators are made by Hewlett Pack-ard, Casio, and Sharp

You can also use computer software such

as Graphing Calculator by Paciic Tech

(www.paciict.com) to perform many of these

functions, as well as apps for phones and

tablets, like Quick Graph (Colombiamug) or

Math-Studio (Pomegranate Apps) Similar

functionality is available using a web interface

Calculators, Computers, and

Other Graphing Devices

The CAS icon is reserved for problems in which the full resources of

a computer algebra system (CAS) are required A CAS is capable of

doing mathematics (like solving equations, computing derivatives or

integrals) symbolically rather than just numerically.

Examples of well-established computer algebra systems are the puter software packages Maple and Mathematica The WolframAlpha website uses the Mathematica engine to provide CAS functionality via the Web

com-Many handheld graphing calculators have CAS capabilities, such

as the TI-89 and TI-Nspire CX CAS from Texas Instruments Some tablet and smartphone apps also provide these capabilities, such as the previously mentioned MathStudio

the use of any of the resources we have mentioned

Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry The fol-lowing tests are intended to diagnose weaknesses that you might have in these areas After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided

6 Rationalize the expression and simplify.

(c) s22, 0d ø s1, `d (d) s1, 7d (e) s21, 4g

If you had dificulty with these problems, you may wish to consult the

Review of Algebra on the website www.stewartcalculus.com.

ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY

(a) has slope 23

(b) is parallel to the x-axis (c) is parallel to the y-axis (d) is parallel to the line 2x 2 4y − 3

(a) Find the slope of the line that contains A and B.

(b) Find an equation of the line that passes through A and B What are the intercepts? (c) Find the midpoint of the segment AB.

(d) Find the length of the segment AB.

(e) Find an equation of the perpendicular bisector of AB.

(f) Find an equation of the circle for which AB is a diameter.

(a) State the value of fs21d.

(b) Estimate the value of fs2d.

(c) For what values of x is fsxd − 2?

(d) Estimate the values of x such that f sxd − 0.

(e) State the domain and range of f.

, evaluate the difference quotient f s2 1 hd 2 f s2d

h and simplify your answer.

FIGURE FOR PROBLEM 1

ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS

4 (a) Relect about the x-axis

(b) Stretch vertically by a factor of 2, then shift 1 unit

downward

(c) Shift 3 units to the right and 2 units upward

x 0

(a)

1 1

y

(b)

x 0 1 _1

(c) y

x 0

(2, 3)

y

(d)

x 0 4

y

(h)

x 0 1

1

6 (a) 23, 3 (b) y

x 0 _1 1

5 Express the lengths a and b in the igure in terms of .

1 1 tan 2

x − sin 2x

a

¨

b

24

FIGURE FOR PROBLEM 5

If you had dificulty with these problems, you should look at Appendix D of this book

If you had dificulty with these problems, you should look at sections 1.1–1.3 of this book

ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY

A Preview of Calculus

CAlCulus Is FunDAmEntAlly DIFFErEnt From the mathematics that you have studied

previ-ously: calculus is less static and more dynamic It is concerned with change and motion; it deals with quantities that approach other quantities For that reason it may be useful to have an overview

of the subject before beginning its intensive study Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety

of problems

The area Problem

The origins of calculus go back at least 2500 years to the ancient Greeks, who found

areas using the “method of exhaustion.” They knew how to ind the area A of any

poly-gon by dividing it into triangles as in Figure 1 and adding the areas of these triangles

It is a much more dificult problem to ind the area of a curved igure The Greek method of exhaustion was to inscribe polygons in the igure and circumscribe polygons about the igure and then let the number of sides of the polygons increase Figure 2 illus-trates this process for the special case of a circle with inscribed regular polygons

A¶

Aß A∞

A¢

A£

Let A n be the area of the inscribed polygon with n sides As n increases, it appears that

A n becomes closer and closer to the area of the circle We say that the area of the circle

is the limit of the areas of the inscribed polygons, and we write

We will use a similar idea in Chapter 5 to ind areas of regions of the type shown in

Figure 3 We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums

of areas of rectangles

1 n

1 2

3 4

The area problem is the central problem in the branch of calculus called integral

cal-culus The techniques that we will develop in Chapter 5 for inding areas will also enable

us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out

of a tank

The Tangent Problem

Consider the problem of trying to ind an equation of the tangent line t to a curve with equation y − f sxd at a given point P (We will give a precise deinition of a tangent line in

TEC In the Preview Visual, you

can see how areas of inscribed and

circumscribed polygons approximate

the area of a circle.

Chapter 2 For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can ind the equation of t if we know its slope m The problem is that we need two points to compute the slope and we know only one point, P, on t To get around the problem we irst ind an approximation

to m by taking a nearby point Q on the curve and computing the slope m PQ of the secant

line PQ From Figure 6 we see that

x 2 a

Now imagine that Q moves along the curve toward P as in Figure 7 You can see that

the secant line rotates and approaches the tangent line as its limiting position This means

that the slope m PQ of the secant line becomes closer and closer to the slope m of the

tan-gent line We write

Speciic examples of this procedure will be given in Chapter 2

The tangent problem has given rise to the branch of calculus called differential calcu-

lus, which was not invented until more than 2000 years after integral calculus The main

ideas behind differential calculus are due to the French mathematician Pierre mat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the Ger-man mathematician Gottfried Leibniz (1646–1716)

Fer-The two branches of calculus and their chief problems, the area problem and the gent problem, appear to be very different, but it turns out that there is a very close con-nection between them The tangent problem and the area problem are inverse problems

tan-in a sense that will be described tan-in Chapter 5

velocity

When we look at the speedometer of a car and read that the car is traveling at 48 miyh, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 mi But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 miyh?

In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in feet) at l-second intervals as in the following chart:

P

Q t

As a irst step toward inding the velocity after 2 seconds have elapsed, we ind the

aver-age velocity during the time interval 2 < t < 4:

average velocity − change in position

time elapsed

4 2 2 − 16.5 ftys

Similarly, the average velocity in the time interval 2 < t < 3 is

Time interval f2, 3g f2, 2.5g f2, 2.4g f2, 2.3g f2, 2.2g f2, 2.1g Average velocity sftysd 15.0 13.6 12.4 11.5 10.8 10.2

The average velocities over successively smaller intervals appear to be getting closer to

a number near 10, and so we expect that the velocity at exactly t − 2 is about 10 ftys In

Chapter 2 we will deine the instantaneous velocity of a moving object as the limiting value

of the average velocities over smaller and smaller time intervals

In Figure 8 we show a graphical representation of the motion of the car by plotting the

distance traveled as a function of time If we write d − f std, then f std is the number of feet traveled after t seconds The average velocity in the time interval f2, tg is

average velocity − change in position

f std 2 f s2d

t2 2

which is the same as the slope of the secant line PQ in Figure 8 The velocity v when

t − 2 is the limiting value of this average velocity as t approaches 2; that is,

v− lim

t l 2

f std 2 f s2d

t2 2and we recognize from Equation 2 that this is the same as the slope of the tangent line

Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences.

The limit of a sequence

In the ifth century bc the Greek philosopher Zeno of Elea posed four problems, now

known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning

space and time that were held in his day Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start Zeno argued, as fol- lows, that Achilles could never pass the tortoise: Suppose that Achil les starts at position

a1 and the tortoise starts at position t1 (See Figure 9.) When Achilles reaches the point

a2− t1, the tortoise is farther ahead at position t2 When Achilles reaches a3− t2, the tor-

toise is at t3 This process continues indeinitely and so it appears that the tortoise will always be ahead! But this deies common sense

Achilles tortoise

.

.

One way of explaining this paradox is with the idea of a sequence The successive

posi-tions of Achilles sa1, a2, a3, d or the successive positions of the tortoise st1, t2, t3, d form what is known as a sequence

In general, a sequence hanj is a set of numbers written in a deinite order For instance, the sequence

terms of the sequence a n− 1yn are becoming closer and closer to 0 as n increases In

fact, we can ind terms as small as we please by making n large enough We say that the

limit of the sequence is 0, and we indicate this by writing

The concept of the limit of a sequence occurs whenever we use the decimal tation of a real number For instance, if

nl ` a n −  The terms in this sequence are rational approximations to .

Let’s return to Zeno’s paradox The successive positions of Achilles and the tortoise form sequences han j and ht n j, where a n , t n for all n It can be shown that both sequences

have the same limit:

lim

n l`a n − p − lim

n l`t n

It is precisely at this point p that Achilles overtakes the tortoise.

The sum of a series

Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall In order to do so, he would irst have to

go half the distance, then half the remaining distance, and then again half of what still remains This process can always be continued and can never be ended.” (See Figure 11.)

1 2

1 4

1 8 1 16

Of course, we know that the man can actually reach the wall, so this suggests that haps the total distance can be expressed as the sum of ininitely many smaller distances

Zeno was arguing that it doesn’t make sense to add ininitely many numbers together But there are other situations in which we implicitly use ininite sums For instance, in decimal notation, the symbol 0.3 − 0.3333 means

13

More generally, if d n denotes the nth digit in the decimal representation of a number, then

we must deine carefully what the sum of an ininite series is

Returning to the series in Equation 3, we denote by s n the sum of the irst n terms of

the series Thus

s10−1

2 1141 ∙ ∙ ∙ 110241 < 0.99902344f

to 1 In fact, it can be shown that by taking n large enough (that is, by adding suficiently many terms of the series), we can make the partial sum s n as close as we please to the num- ber 1 It therefore seems reasonable to say that the sum of the ininite series is 1 and to write

In other words, the reason the sum of the series is 1 is that

After Sir Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in cal-culating life insurance premiums, and in a great variety of other areas We will explore some of these uses of calculus in this book

In order to convey a sense of the power of the subject, we end this preview with a list

of some of the questions that you will be able to answer using calculus:

1 How can we explain the fact, illustrated in Figure 12, that the angle of elevation

from an observer up to the highest point in a rainbow is 42°? (See page 285.)

2 How can we explain the shapes of cans on supermarket shelves? (See page 343.)

3 Where is the best place to sit in a movie theater? (See page 465.)

4 How can we design a roller coaster for a smooth ride? (See page 182.)

5 How far away from an airport should a pilot start descent? (See page 208.)

6 How can we it curves together to design shapes to represent letters on a laser

printer? (See page 657.)

7 How can we estimate the number of workers that were needed to build the Great

Pyramid of Khufu in ancient Egypt? (See page 460.)

8 Where should an inielder position himself to catch a baseball thrown by an

outielder and relay it to home plate? (See page 465.)

9 Does a ball thrown upward take longer to reach its maximum height or to fall

back to its original height? (See page 609.)

10 How can we explain the fact that planets and satellites move in elliptical orbits?

(See page 868.)

11 How can we distribute water low among turbines at a hydroelectric station so

as to maximize the total energy production? (See page 980.)

12 If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which

of them reaches the bottom irst? (See page 1052.)

rays from sun